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diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex index 64a098e..dd862ef 100644 --- a/chapter04/content_ch04.tex +++ b/chapter04/content_ch04.tex @@ -1118,13 +1118,259 @@ It can be shown that the vectors $\cmplxvect{u}_k$ are orthogonal. They form an The basis of the frequency samples of a \ac{DFT} are orthogonal. \end{fact} -\subsection{Windowing Non-Periodic Signals} +\subsection{Processing of Non-Periodic Signals} -\todo{Windowing and Periodic continuation} +In practical digital systems, signals are non-periodic. A non-periodic sequence $\underline{x}[n]$ can be fully transformed by the \ac{DTFT}. +\begin{equation} + \underline{x}[n] \TransformHoriz \mathcal{F}_{\text{DTFT}}\left\{\underline{x}[n]\right\} = \underline{X}_{\frac{2\pi}{T_S}}\left(e^{j T_S \omega}\right) +\end{equation} + +However, it is not feasible to implement a \ac{DTFT}, because the sums and integrals run over an indefinite interval. Therefore, the transforms are approximated by the \ac{DFT}. + +\subsubsection{Windowing and Periodic Continuation} + +A sequence of length $N$ is taken out of $\underline{x}[n]$: +\begin{equation} + \underline{\tilde{x}}_N[n] = \left[\underline{x}[m], \underline{x}[m+1], \underline{x}[m+1], \dots, \underline{x}[m+N-1]\right] +\end{equation} +The act of extracting $N$ subsequent samples out of $\underline{x}[n]$ is called \index{windowing} \textbf{windowing}. To illustrate this, imagine that you watch a sequence moving to the left through a window. The movement to the left is the time advance. Because of the window, you see a certain extract of $\underline{x}[n]$ only at one time. + +\todo{Window illustration} + +This sequence is repeated indefinitely (\emph{periodic continuation}): +\begin{equation} + \underline{x}_N[n] = \underline{\tilde{x}}_N[n \mod N] +\end{equation} +Thus, $\underline{x}_N[n]$ is periodic: +\begin{equation} + \underline{x}_N[n] = \underline{x}_N[n + q N] \qquad \forall \; q \in \mathbb{Z} +\end{equation} + +Now, $\underline{x}_N[n]$ can be transformed to the frequency-domain using the \ac{DFT}. +\begin{equation} + \underline{x}_N[n] \TransformHoriz \mathcal{F}_{\text{DFT}}\left\{\underline{x}[n]\right\} = \underline{X}[k] +\end{equation} + +The periodic continuation is just a theoretical explanation. In practise, it is enough to calculate the \ac{DFT} over $N$ subsequent samples out of $\underline{x}[n]$. +\begin{equation} + \begin{split} + \underline{X}_{\frac{2\pi}{T_S}}\left(e^{j T_S \omega}\right) &\approx \begin{cases} + \left.\underline{X}[k]\right|_{k = N \frac{T_S}{2\pi} \omega} & \quad \text{if } \left(N \frac{T_S}{2\pi} \omega\right) \in \mathbb{Z}, \\ + 0 & \quad \text{if } \left(N \frac{T_S}{2\pi} \omega\right) \notin \mathbb{Z}. + \end{cases} \\ + \underline{X}_{\frac{2\pi}{T_S}}\left(e^{j 2 \pi \frac{k}{N}}\right) &\approx \underline{X}[k] = \sum\limits_{n=0}^{N-1} \underline{x}_N[n] e^{-j 2 \pi \frac{k}{N} n} + \end{split} + \label{eq:ch04:dtft_sampling} +\end{equation} + +Windowing and periodic continuation is, in fact, a sampling of the \ac{DTFT} (see \eqref{eq:ch04:dtft_sampling}). + +\subsubsection{Window Functions} + +The widowing of $\underline{x}[n]$ to obtain $\underline{x}_N[n]$ as discussed before did not change the values of $\underline{x}[n]$. This can be expressed as a multiplication of the original sequence with the rectangular function: +\begin{equation} + \underline{x}_W[n] = \underline{x}_N[n] \cdot \underbrace{w_{rect,N}[n]}_{\text{Rectangular window}} +\end{equation} +where +\begin{equation} + w_{rect,N}[n] = \begin{cases} + 1 &\quad \text{if } \; 0 \geq n < N, \\ + 0 &\quad \text{else}. + \end{cases} +\end{equation} -\todo{Window Filters} +$w[n]$ is called \index{window function} \textbf{window function}. +\begin{definition}{Window function} + A \index{window function} \textbf{window function} $w[n]$ is applied to the periodically continued sequence $\underline{x}_N[n]$ by multiplication: + \begin{equation} + \underline{x}_N[n] \equiv \underline{x}[n] \cdot w[n] + \end{equation} +\end{definition} + +\subsubsection{Spectral Leakage} + +In the frequency-domain, the multiplication becomes a convolution: +\begin{equation} + \begin{split} + \underline{X}_W[k] &= \mathcal{F}_{\text{DFT}}\left\{\underline{x}_W[n]\right\} \\ + &= \mathcal{F}_{\text{DFT}}\left\{\underline{x}_N[n] \cdot w[n]\right\} \\ + &= \underline{X}_N[k] * \underline{W}[k] \\ + &= \sum\limits_{l=0}^{N-1} \underline{X}_N[l] * \underline{W}[k-l] + \end{split} +\end{equation} + +This has some implications: +\begin{itemize} + \item Each frequency component in $\underline{X}_N[k]$ creates frequency components in \underline{all} $\underline{X}_W[k]$. + \item A mono-chromatic signal, which would be non-zero at only one $k$ and zero everywhere else, has non-zero components everywhere in $\underline{X}_W[k]$. + \item The non-zero frequency components produced by the convolution show up as noise and \underline{decrease the \ac{SNR}}. +\end{itemize} +This effect is called \index{spectral leakage} \textbf{spectral leakage}. + +\begin{figure}[H] + \centering + + \subfloat[Original analogue signal]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.35\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x(t)$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=0, + xmax=4.5, + ymin=-1.2, + ymax=1.2, + %xtick={-1, -0.5, 0, 0.5, 1}, + %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, + %ytick={0}, + ] + \addplot[red, thick, smooth, domain=0:4, samples=50] plot(\x, {cos(deg(pi*\x))}); + \end{axis} + \end{tikzpicture} + } + \hfill + \subfloat[Window function in the time-domain]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.35\linewidth, + scale only axis, + xlabel={$n$}, + ylabel={$w[n]$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-0.5, + xmax=8.5, + ymin=0, + ymax=1.2, + %xtick={-1, -0.5, 0, 0.5, 1}, + %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, + %ytick={0}, + ] + \addplot[blue, thick] coordinates {(-0.5,0) (0,0)}; + \addplot[blue, thick] coordinates {(0,1) (8,1)}; + \addplot[blue, thick] coordinates {(8,0) (8.5,0)}; + + \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)}; + \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)}; + \end{axis} + \end{tikzpicture} + } + + \subfloat[Amplitude spectrum of the periodically continued signal $\underline{X}_N[k]$]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.35\linewidth, + scale only axis, + xlabel={$k$}, + ylabel={$\underline{X}_N[k]$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=0, + xmax=4.5, + ymin=-1.2, + ymax=1.2, + %xtick={-1, -0.5, 0, 0.5, 1}, + %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, + %ytick={0}, + ] + \addplot[red, thick, smooth, domain=0:4, samples=50] plot(\x, {cos(deg(pi*\x))}); + \end{axis} + \end{tikzpicture} + } + \hfill + \subfloat[Amplitude spectrum of the window ]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.35\linewidth, + scale only axis, + xlabel={$k$}, + ylabel={$\underline{W}[k]$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-4.5, + xmax=4.5, + ymin=0, + ymax=1.2, + %xtick={-1, -0.5, 0, 0.5, 1}, + %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$}, + %ytick={0}, + ] + \addplot[blue, thick] coordinates {(-0.5,0) (0,0)}; + \addplot[blue, thick] coordinates {(0,1) (8,1)}; + \addplot[blue, thick] coordinates {(8,0) (8.5,0)}; + + \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)}; + \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)}; + \end{axis} + \end{tikzpicture} + } + + \caption{Spectral leakage of a mono-chromatic signal} +\end{figure} + +\begin{landscape} + \subsubsection{Some Window Functions} +\end{landscape} -\todo{Spectral Leakage} \section{Analogies Of Time-Continuous and Time-Discrete Signals and Systems} @@ -1368,6 +1614,8 @@ Parseval's theorem for discrete systems: \underline{\mathrm{S}}_{E,xx}[k] = \left|\underline{X}[k]\right|^2 \end{equation} +\todo{PSD} + %\subsection{Noise} \section{Digital Signals and Systems} @@ -1411,15 +1659,24 @@ Quantization is the process of height={0.25\textheight}, width=0.6\linewidth, scale only axis, - xlabel={$t$}, + xlabel={$n$}, ylabel={$x$}, %grid style={line width=.6pt, color=lightgray}, %grid=both, xmajorgrids=false, ymajorgrids=true, grid style={color=lightgray, dashed}, - axis lines=left, legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, xmin=0, xmax=7, ymin=0, @@ -1460,19 +1717,167 @@ The finite set of discrete numbers has the length $K$. \item Usually, $K$ is a power of $2$. $K = 2^M$. $M$ is the number of bits. \end{itemize} -\subsubsection{Linear Mapping} +\subsubsection{Linear Rounding Quantizer} + +Rounding is a common model for quantization. \textit{Remark:} There are other methods like dead-zone quantizers, which are not subject to this lecture. Refer to literature about electronics to get a comprehensive overview. The most common implementations distribute the $K$ discrete values equally between an interval of the continuous values $[\underline{\hat{X}}_L, \underline{\hat{X}}_H]$. So, the discrete values are spaced by \begin{equation} - \Delta \underline{\hat{X}} = \frac{\underline{\hat{X}}_H - \underline{\hat{X}}_L}{K} + \Delta \underline{\hat{X}} = \frac{\underline{\hat{X}}_H - \underline{\hat{X}}_L}{K - 1} \qquad \forall \, K \geq 1 \end{equation} -This is called \index{linear mapping} \textbf{linear mapping}. +The mapping between $\underline{x}[n]$ and $\underline{x}_Q[n]$ is therefore \emph{linear}. + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.4\textheight}, + width=0.8\linewidth, + scale only axis, + xlabel={$x[n]$}, + ylabel={$x_Q[n]$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + xmajorgrids=false, + ymajorgrids=true, + grid style={color=lightgray, dashed}, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xticklabel style={rotate=90, anchor=east}, + xmin=0, + xmax=9, + ymin=0, + ymax=9, + xtick={0, 1, ..., 8}, + xticklabels={$0$, $\hat{X}_L$, $\hat{X}_L + \Delta \hat{X}$, $\hat{X}_L + 2 \Delta \hat{X}$, $\hat{X}_L + 3 \Delta \hat{X}$, $\hat{X}_L + 4 \Delta \hat{X}$, $\hat{X}_L + 5 \Delta \hat{X}$, $\hat{X}_L + 6 \Delta \hat{X}$, $\hat{X}_H$}, + ytick={0, 1, ..., 8}, + yticklabels={$0$, $\hat{X}_L$, $\hat{X}_L + \Delta \hat{X}$, $\hat{X}_L + 2 \Delta \hat{X}$, $\hat{X}_L + 3 \Delta \hat{X}$, $\hat{X}_L + 4 \Delta \hat{X}$, $\hat{X}_L + 5 \Delta \hat{X}$, $\hat{X}_L + 6 \Delta \hat{X}$, $\hat{X}_H$}, + ] + \addplot[red, thick] coordinates {(0, 1) (1.5, 1)}; + \addplot[red, thick, dashed] coordinates {(1.5, 1) (1.5, 2)}; + \pgfplotsinvokeforeach{2, 3, ..., 7}{ + \addplot[red, thick] coordinates {({#1-0.5}, #1) ({#1+0.5}, #1)}; + \addplot[red, thick, dashed] coordinates {({#1+0.5}, #1) ({#1+0.5}, {#1+1})}; + } + \addplot[red, thick] coordinates {(7.5, 8) (9, 8)}; + \end{axis} + \end{tikzpicture} + \caption[A rounding quantizer with linear mapping and a set of $K = 8$ discrete values]{A rounding quantizer with linear mapping and a set of $K = 8$ discrete values. All values $x[n] < \hat{X}_L$ are mapped to $\hat{X}_L$ and all values $x[n] > \hat{X}_H$ are mapped to $\hat{X}_H$.} + \label{fig:ch04:linear_ammping} +\end{figure} + + +\begin{figure}[H] + \centering + \begin{circuitikz} + \node[op amp] (Cmp1){}; + \node[op amp, below=1.5cm of Cmp1] (Cmp2){}; + \node[below=1cm of Cmp2] (Cmpx){$\vdots$}; + \node[op amp, below=1cm of Cmpx] (Cmpn){}; + + \draw (Cmp1.-) to[short] ++(-1cm, 0); + \draw (Cmp2.-) to[short] ++(-1cm, 0); + \draw (Cmpn.-) to[short] ++(-1cm, 0); + \draw ([shift={(-1cm,1.5cm)}] Cmp1.-) node[vcc]{$U_{ref}$} to[R,l=$R$,-*] ([shift={(-1cm,0)}] Cmp1.-) + to[R,l=$R$,-*] ([shift={(-1cm,0)}] Cmp2.-) + to[short] ++(0,-1.5cm) + to[open] ([shift={(-1cm,1.5cm)}] Cmpn.-) + to[short,-*] ([shift={(-1cm,0)}] Cmpn.-) + to[short] ([shift={(-1cm,-1.5cm)}] Cmpn.-) + to[R,l=$R$] ([shift={(-1cm,-3cm)}] Cmpn.-) node[rground]{}; + + \draw ([shift={(-4cm,0)}] Cmp1.+) node[left]{$u_{in}(t)$} to[short,o-] (Cmp1.+); + \draw ([shift={(-2cm,0)}] Cmp1.+) to[short,*-] ([shift={(-2cm,0)}] Cmp2.+) to[short,-] (Cmp2.+); + \draw ([shift={(-2cm,0)}] Cmp2.+) to[short,*-] ++(0,-1cm); + \draw ([shift={(-2cm,2cm)}] Cmpn.+) to[short,-] ([shift={(-2cm,0)}] Cmpn.+) to[short,-] (Cmpn.+); + + \node[below] at([shift={(-1cm,-1.5cm)}] Cmp2.-){$\vdots$}; + \node[below] at([shift={(-2cm,-1cm)}] Cmp2.+){$\vdots$}; + + \draw (Cmpn.out) to[short,-o] ++(2cm,0) node[right,align=left]{$1 \quad \text{if } u_{in}(t) \geq \frac{1}{K-1} U_{ref}$}; + \draw (Cmp2.out) to[short,-o] ++(2cm,0) node[right,align=left]{$1 \quad \text{if } u_{in}(t) \geq \frac{K-3}{K-1} U_{ref}$}; + \draw (Cmp1.out) to[short,-o] ++(2cm,0) node[right,align=left]{$1 \quad \text{if } u_{in}(t) \geq \frac{K-2}{K-1} U_{ref}$}; + \end{circuitikz} + \caption[An example quantizer circuit implementing the rounding quantization using comparators]{An example quantizer circuit implementing the rounding quantization using comparators. $K$ resistors $R$ and $K-1$ linearly quantize $u_{in}$ into $K$ discrete values. The range is \SI{0}{V} to $U_{ref}$.} +\end{figure} + +\textit{Remark:} There are other mappings like logarithmic mapping. However, this lecture only considers linear mapping. -\todo{step map} +\subsection{Pulse-Code Modulation} -\todo{Example circuit} +One piece of the \ac{ADC} is missing. So far, we a have quantized, discrete signal $\underline{x}_Q[n]$. + +\begin{itemize} + \item To be processed by a computer, the values of $\underline{x}_Q[n]$ must be coded in way a computer can handle. + \item A computer processes signals with two discrete states $0$ and $1$. These binary signals are called \index{bit} \textbf{bits}. + \item Multiple bits are grouped to symbols. + \begin{itemize} + \item \index{nibble} 1 nibble: 4 bits + \item \index{byte} 1 byte: 8 bits + \item \index{word} 1 word: natural data size of a computer architecture (most commonly 32 bits or 64 bits) + \end{itemize} +\end{itemize} + +The discrete values of $\underline{x}_Q[n]$ are coded as computer readable symbols. This mapping is called \index{pulse-code modulation} \textbf{\ac{PCM}}. + +\begin{table}[H] + \centering + \caption[Example PCM for $K = 8$]{Example \ac{PCM} for $K = 8$} + \begin{tabular}{|l|r|r|} + \hline + $\underline{x}_Q[n]$ & Data & Data (binary) \\ + \hline + \hline + $\hat{X}_L$ & 0 & 000 \\ + \hline + $\hat{X}_L + \Delta \hat{X}$ & 1 & 001 \\ + \hline + $\hat{X}_L + 2 \Delta \hat{X}$ & 2 & 010 \\ + \hline + $\hat{X}_L + 3 \Delta \hat{X}$ & 3 & 011 \\ + \hline + $\hat{X}_L + 4 \Delta \hat{X}$ & 4 & 100 \\ + \hline + $\hat{X}_L + 5 \Delta \hat{X}$ & 5 & 101 \\ + \hline + $\hat{X}_L + 6 \Delta \hat{X}$ & 6 & 110 \\ + \hline + $\hat{X}_H$ & 7 & 111 \\ + \hline + \end{tabular} +\end{table} -\textit{Remark:} There are other mapping like logarithmic mapping. However, this lecture only considers linear mapping. +The complete processing chain of the \ac{PCM}, implemented by an \ac{ADC}, be completed to: +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{tikzpicture} + \node[draw, block] (Sampler){Sampling}; + \node[draw, block, right=1cm of Sampler] (Quantizer){Quantizer}; + \node[draw, block, right=1cm of Quantizer] (Coder){Coder}; + + \draw[o->] ([xshift=-1.5cm] Sampler.west) node[left,align=right]{Time-continuous,\\ value-continuous\\ (analog) signal\\ $\underline{x}(t)$} -- (Sampler.west); + \draw[->] (Sampler.east) -- (Quantizer.west); + \draw[->] (Quantizer.east) -- (Coder.west); + \draw[->] (Coder.east) -- ++(1.5cm,0) node[right,align=left]{Time-discrete,\\ value-discrete,\\ and coded\\ (digital) data}; + + \draw[o->] ([yshift=1cm] Sampler.north) node[above, align=center]{Sampling clock\\ $\Sha_{T_S}(t)$} -- (Sampler.north); + + \draw[decorate, decoration={brace, amplitude=3mm, mirror}] ([yshift=-5mm] Sampler.south west) -- ([yshift=-5mm] Coder.south east) node[midway, below, yshift=-3mm]{\textbf{\acf{ADC}}}; + \end{tikzpicture} + \end{adjustbox} + \caption[Generating a PCM signal from an analogue signal $\underline{x}(t)$ using an ADC]{Generating a \ac{PCM} signal from an analogue signal $\underline{x}(t)$ using an \ac{ADC}} + \label{fig:ch04:signals_to_data} +\end{figure} \subsection{Quantization Noise} @@ -1492,17 +1897,334 @@ Furthermore, the quantized values differ from their original value due to roundi The error is bounded to \begin{equation} - \left|\underline{e}[n]\right| \leq \frac{1}{2} \Delta \underline{\hat{X}} + \left|\underline{e}[n]\right| \leq \frac{1}{2} \left|\Delta \underline{\hat{X}}\right| \end{equation} \end{definition} -\todo{Quantization Noise Floor} +\subsubsection{Quantization Error of PCM Data} -\todo{Dynamic Range, dBFS} +\ac{PCM} data is usually coded as symbols consisting of $B$ (binary) bits. +\begin{itemize} + \item Subsequent symbols have a spacing of $1$ (unity spacing). + \item The \ac{PCM} normalizes the value spacing of the quantizer $\underline{\hat{X}}$ to unity spacing $1$. + \item A corollary is that the quantization error $\left|\underline{\tilde{e}}[n]\right| = \frac{1}{2}$. +\end{itemize} +For example if the \ac{PCM} value is $\left(10010100\right)_2$, its real value is between $\left(10010011\right)_2$ (equivalent to $\left(10010100\right)_2 - 0.5$) and $\left(10010110\right)_2$ (equivalent to $\left(10010100\right)_2 + 0.5$). + +\begin{itemize} + \item The quantization error of a \ac{PCM} signal resides in \ac{LSB}. + \item The quantization error is uniformly distributed between $\SI{-1/2}{LSB} = -2^{-B}$ and $\SI{+1/2}{LSB} = +2^{-B}$, where $B$ is the number of bits. +\end{itemize} + +\begin{excursus}{The \ac{LSB}} + A binary number can be decomposed: + \begin{equation} + \left(10010101\right)_2 = 1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 + \end{equation}% + \nomenclature[Nb]{$\left(\cdot\right)_2$}{Binary number}% + \nomenclature[Nh]{$\left(\cdot\right)_{16}$}{Hexadecimal number}% + \nomenclature[Nd]{$\left(\cdot\right)_{10}$}{Explicit decimal number} + + The \index{least significant bit} \textbf{\ac{LSB}} is the digit with the exponent $2^0$. +\end{excursus} + +\todo{Plot error} + +The quantization error superimposes the original signal and is therefore noise -- the \index{quantization noise} \textbf{quantization noise}. + +The \emph{quantization noise power} of the error, which is distributed between $-2^{-B}$ and $+2^{-B}$, is: +\begin{equation} + P_{Q,N} = \hat{X}_H^2 2^{B-1} \int\limits_{-2^{-B}}^{+2^{-B}} e^2 \, \mathrm{d} e = \frac{\hat{X}_H^2}{3 \cdot 4^B} +\end{equation} +where $\hat{X}_H$ is the highest discrete value of the quantizer for input signals $x[n]$. + +\begin{definition}{The \acf{SQNR}} + A \ac{SNR} which is referenced to the quantization noise is called \index{signal-to-qunatization-noise ratio} \textbf{\acf{SQNR}}. + \begin{equation} + \mathrm{SQNR} = \frac{P_{Q,S}}{P_{Q,N}} = \frac{P_{Q,S} \cdot 3 \cdot 4^B}{\hat{X}_H^2} + \end{equation} + where $\hat{X}_H$ is the highest discrete value of the quantizer for input signals $x[n]$. + Or in the logarithmic scale: + \begin{equation} + \begin{split} + L_{\mathrm{SQNR}} &= \SI{10}{dB} \log_{10}\left(\frac{P_{Q,S}}{P_{Q,N}}\right) \\ + &= \SI{10}{dB} \log_{10}\left(\frac{P_{Q,S} \cdot 3 \cdot 4^B}{\hat{X}_H^2}\right) \\ + &= \SI{10}{dB} \log_{10}\left(\frac{P_{Q,S}}{\hat{X}_H^2}\right) + \SI{4.77}{dB} + B \cdot \SI{6.02}{dB} + \end{split} + \end{equation} +\end{definition} + +%TODO Is this correct? +%As a rule of thumb, +%\begin{itemize} +% \item The overall \ac{SNR} is the \ac{SQNR} for \underline{strong} input signals. The quantization noise dominates the thermal noise. +% \item The overall \ac{SNR} is the signal-to-thermal-noise ratio for \underline{weak} input signals. The quantization noise is dominated by the thermal noise. +%\end{itemize} + +\subsubsection{Dynamic Range} + +The maximum range of a quantizer is $\hat{X}_H - \hat{X}_L$. $\hat{X}_L$ is zero is most applications. $\hat{X}_H$ is the highest discrete value of the quantizer for input signals $x[n]$. + +An important characteristic of an \ac{ADC} is its \emph{resolution}. It is approximately $\frac{\hat{X}_H}{2^B}$. + +Let's consider the \ac{SQNR} for a sine wave at maximum amplitude $\hat{X}_H$. The power of the quantized sine wave is: +\begin{equation} + P_{Q,S} = \frac{1}{2} \hat{X}_H^2 +\end{equation} +The \ac{SQNR} is: +\begin{equation} + \left.L_{\mathrm{SQNR}}\right|_{\text{Sine}} = \SI{10}{dB} \log_{10}\left(\frac{1}{2}\right) + \SI{4.77}{dB} + B \cdot \SI{6.02}{dB} = \SI{1.761}{dB} + B \cdot \SI{6.02}{dB} +\end{equation} + +The \ac{SQNR} of a sine wave is the \index{dynamic range} \textbf{dynamic range} of an \ac{ADC}. +\begin{definition}{Dynamic range of an \ac{ADC}} + \begin{equation} + L_{\mathrm{DR}} = \SI{1.761}{dB} + B \cdot \SI{6.02}{dB} + \end{equation} +\end{definition} + +The dynamic range increases, when more bits $B$ are added. However, the number of bits $B$ correlates with the complexity of the \ac{ADC} hardware. Therefore, a trade-off between sufficient dynamic range and reasonable hardware complexity must be chosen. Typical bit numbers: +\begin{itemize} + \item $B = 8$ to $B = 10$ for \ac{RF} applications with high sampling rates ($> \SI{1}{MHz}$), + \item $B = 16$ or $B = 24$ for audio applications with medium sampling rates ($\approx \SI{44}{kHz}$). +\end{itemize} + +%\begin{excursus}{\si{dBFS}} +% When talking about signals in relation to an \ac{ADC}, they are referenced. +%\end{excursus} + +%\subsubsection{Quantization Noise Floor} +% +%Analogous to the thermal noise floor, \subsection{Time Recovery} -\subsection{Practical Issues} +For wideband \ac{RF} applications, the sampling rate is chosen close to the maximum frequency of the signal, so that the Shannon-Nyquist sampling theorem can be fulfilled. + +The sampling rate $f_S$ is not the only important parameter. The time instance at which the sample is taken must fit, too. Thus, both period and phase of the Dirac comb $\Sha_{T_S}(t)$ used by the sampler must fit. + +The problem with a non-optimal sampling phase is depicted below. The difference between the current sample timing and the optimal sample timing is the \index{timing error} \textbf{timing error}. + +\begin{figure}[H] + \centering + \subfloat[Optimal sample timing]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.20\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=outer north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=0, + xmax=4.5, + ymin=-1.2, + ymax=1.2, + xtick={0, 1, ..., 4}, + ytick={-1, -0.5, ..., 1} + ] + \addplot[red, smooth, domain=0:4, samples=50] plot (\x, {cos(deg(pi*\x))}); + \addlegendentry{Original signal}; + \addplot[green, dashed, smooth, domain=0:4, samples=50] plot (\x, {cos(deg(pi*\x))}); + \addlegendentry{Sampled signal}; + \addplot[blue, only marks, mark=o] coordinates {(0, 1) (1, -1) (2, 1) (3, -1) (4, 1)}; + \addlegendentry{Sample times}; + \end{axis} + \end{tikzpicture} + } + + \subfloat[Non-optimal sample timing]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.20\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=outer north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=0, + xmax=4.5, + ymin=-1.7, + ymax=1.2, + xtick={0, 1, ..., 4}, + ytick={-1, -0.5, ..., 1} + ] + \addplot[red, smooth, domain=0:4, samples=50] plot (\x, {cos(deg(pi*\x))}); + \addlegendentry{Original signal}; + \addplot[green, dashed, smooth, domain=0:4, samples=50] plot (\x, {0.707*cos(deg(pi*\x-(pi/4)))}); + \addlegendentry{Sampled signal}; + \addplot[blue, only marks, mark=o] coordinates {(0.25, 0.707) (1.25, -0.707) (2.25, 0.707) (3.25, -0.707)}; + \addlegendentry{Sample times}; + + \draw[dashed] (axis cs:2,1.2) -- (axis cs:2,-1.2); + \draw[dashed] (axis cs:2.25,1.2) -- (axis cs:2.25,-1.2); + \draw[<->] (axis cs:2,-1.1) -- node[midway,yshift=-2mm,below,align=center]{Timing error} (axis cs:2.25,-1.1); + \end{axis} + \end{tikzpicture} + } + + \subfloat[Poor sample timing]{ + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.20\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=outer north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=0, + xmax=4.5, + ymin=-1.7, + ymax=1.2, + xtick={0, 1, ..., 4}, + ytick={-1, -0.5, ..., 1} + ] + \addplot[red, smooth, domain=0:4, samples=50] plot (\x, {cos(deg(pi*\x))}); + \addlegendentry{Original signal}; + \addplot[green, dashed] coordinates {(0, 0) (4, 0)}; + \addlegendentry{Sampled signal}; + \addplot[blue, only marks, mark=o] coordinates {(0.5, 0) (1.5, 0) (2.5, 0) (3.5, 0)}; + \addlegendentry{Sample times}; + + \draw[dashed] (axis cs:2,1.2) -- (axis cs:2,-1.2); + \draw[dashed] (axis cs:2.5,1.2) -- (axis cs:2.5,-1.2); + \draw[<->] (axis cs:2,-1.1) -- node[midway,yshift=-2mm,below,align=center]{Timing error} (axis cs:2.5,-1.1); + \end{axis} + \end{tikzpicture} + } + \caption[Different situations of sample timing]{Different situations of sample timing. In this example, the signal frequency is close to half of the sampling rate (maximum allowed frequency according to the Nyquist-Shannon sampling theorem).} + \label{fig:ch04:timing_error} +\end{figure} + +The phase (time offset) of the sampler must be adjusted to obtain an optimal sample timing. This adjustment is called \index{timing recovery} \textbf{timing recovery}. The goal is to reduce the \emph{timing error} to zero. + +There are different approaches to implement the timing recovery: +\begin{itemize} + \item Analogue timing recovery (with forward prediction) + \item Hybrid timing recovery (with closed control loop) + \item Digital timing recovery (with forward prediction or closed control loop) +\end{itemize} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + \node[draw, block] at(0,0) (Sampler){Sampler}; + \node[oscillator, below=of Sampler](Clock){}; + \node[draw, block, left=of Clock] (Filter){Loop filter}; + \node[draw, block, left=of Filter] (Pred){Timing error\\ prediction}; + + \draw[dashed] (Sampler.north) -- ++(0, 2cm) node[below left, align=right]{Analogue\\ domain} node[below right, align=left]{Digital\\ domain}; + \node[right=2mm of Clock, align=left]{Sampling clock}; + + \draw[o->] (-11,0) node[left,align=right]{Input} -- (Sampler.west); + \draw[->] (Sampler.east) -- ++(1.5,0) node[right,align=left]{Further signal\\ processing}; + + \draw[*->] (-10,0) |- (Pred.west); + \draw[->] (Pred.east) -- (Filter.west); + \draw[->] (Filter.east) -- (Clock.west); + \draw[->] (Clock.north) -- (Sampler.south); + \end{circuitikz} + \end{adjustbox} + \caption{Analogue timing recovery (with forward prediction)} +\end{figure} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + \node[draw, block] at(0,0) (Sampler){Sampler}; + \node[oscillator, below=of Sampler](Clock){}; + \node[draw, block, right=of Clock] (Filter){Loop filter}; + \node[draw, block, right=of Filter] (Pred){Timing error\\ estimator}; + + \draw[dashed] (Sampler.north) -- ++(0, 2cm) node[below left, align=right]{Analogue\\ domain} node[below right, align=left]{Digital\\ domain}; + \node[left=2mm of Clock, align=right]{Sampling clock}; + + \draw[o->] (-2.5,0) node[left,align=right]{Input} -- (Sampler.west); + \draw[->] (Sampler.east) -- ++(11,0) node[right,align=left]{Further signal\\ processing}; + + \draw[*->] (10,0) |- (Pred.east); + \draw[->] (Pred.west) -- (Filter.east); + \draw[->] (Filter.west) -- (Clock.east); + \draw[->] (Clock.north) -- (Sampler.south); + \end{circuitikz} + \end{adjustbox} + \caption{Hybrid timing recovery (with closed control loop)} +\end{figure} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + \node[draw, block] at(0,0) (Sampler){Sampler}; + \node[oscillator, below=of Sampler](Clock){}; + \node[draw, block, right=of Sampler] (Resampler){Interpolation\\ and resampling}; + \node[draw, block, below=of Resampler] (Filter){Loop filter}; + \node[draw, block, right=of Filter] (Pred){Timing error\\ estimator}; + + \draw[dashed] (Sampler.north) -- ++(0, 2cm) node[below left, align=right]{Analogue\\ domain} node[below right, align=left]{Digital\\ domain}; + \node[left=2mm of Clock, align=right]{Free-running\\ sampling clock}; + + \draw[o->] (-2.5,0) node[left,align=right]{Input} -- (Sampler.west); + \draw[->] (Sampler.east) -- (Resampler.west); + \draw[->] (Resampler.east) -- ++(6,0) node[right,align=left]{Further signal\\ processing}; + + \draw[*->] (11,0) |- (Pred.east); + \draw[->] (Pred.west) -- (Filter.east); + \draw[->] (Filter.north) -- (Resampler.south); + + \draw[->] (Clock.north) -- (Sampler.south); + \end{circuitikz} + \end{adjustbox} + \caption{Analogue timing recovery (with closed control loop)} +\end{figure} + +There various algorithms to implement the timing recovery, which will not be discussed in detail in this lecture. + +%\subsection{Practical Issues} \phantomsection \addcontentsline{toc}{section}{References} diff --git a/common/acronym.tex b/common/acronym.tex index 605a487..4074f81 100644 --- a/common/acronym.tex +++ b/common/acronym.tex @@ -107,6 +107,7 @@ \acro{OSI}{Open Systems Interconntection} \acro{PAN}{personal area network} \acro{PC}{personal computer} + \acro{PCM}{pulse-code modulation} \acro{PCB}{printed circuit board} \acro{PDF}{probability density function} \acro{PHR}{physical layer header} @@ -139,6 +140,7 @@ \acro{SHF}{super high frequency} \acro{SLF}{super low frequency} \acro{SNR}{signal-to-noise ratio} + \acro{SQNR}{signal-to-qunatization-noise ratio} \acro{SPI}{serial peripheral interface} \acro{TCXO}{temperature-compensated crystal oscillator} \acro{TCP}{Transmission Control Protocol} |